NON-CONSTANT SUM RED-AND-BLACK GAMES WITH BET-DEPENDENT WIN PROBABILITY FUNCTION LAURA PONTIGGIA, University of the Sciences in Philadelphia

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1 To appear n Journal o Appled Probablty June 2007 O-COSTAT SUM RED-AD-BLACK GAMES WITH BET-DEPEDET WI PROBABILITY FUCTIO LAURA POTIGGIA, Unversty o the Scences n Phladelpha Abstract In ths paper we nvestgate a class o -person non-constant sum redand-black games wth bet-dependent wn probablty unctons. We assume that players and a gamblng house are engaged n a game played n stages, where the player s probablty o wnnng at each stage s a uncton o the rato o hs bet to the sum o all players bets. However, at each stage o the game there s a postve probablty that all players lose and the gamblng house wns ther bets. We prove that the wn probablty uncton s super-addtve and t satses st st, then a bold strategy s optmal or all players. Keywords: Stochastc games on-constant sum games Red-and-black AMS 2000 Subject Classcaton: Prmary 91A15 Secondary 91A05; 91A06 1. Introducton One o the most nterestng examples n Dubns and Savage [2] s the amous red-and-black gamblng problem. In the dscrete verson ths gamblng problem can be descrbed as ollows. A player has an ntal nonnegatve nteger ortune x and wants to reach a goal o g unts by bettng at each stage o the game an ntegral amount s not greater than hs current ortune. I hs ntal stake s s x, hs next ortune wll be x + s wth probablty w wth 0 < w < 1, and x s wth probablty w 1 w. And so on untl the goal g s reached or the player goes broke. The problem or the player s to decde how many unts o hs current ortune should be staked at each stage n order to maxmze hs probablty o reachng the goal g. Dubns and Savage [2] showed that the soluton o ths problem depends on the value o the probablty w. In partcular, they showed that n the subar case when w 1/2, an optmal strategy s bold play whch corresponds to always stakng the entre ortune or just what s needed to reach the target, whchever s smaller. In the superar case.e. w 1/2 Ross [6] proved that t s optmal or the player to play tmdly, that s always to stake 1 unt o hs current ortune when he s between 1 and g 1 and 0 otherwse. A gamblng Postal address: 600 South 43rd Street, Phladelpha, PA 19104, USA Emal address: 1

2 2 LAURA POTIGGIA theoretcal proo o Ross s result can be ound n Matra and Sudderth [3]. Pontgga [5] ntroduced two-person red-and-black games, n whch two players hold an ntegral amount o money and they both am to wn the other player s ortune. Chen and Hsau [1] extended the results n Pontgga [5] to two-person red-and-black games wth bet-dependent wn probablty unctons. At the present tme we are not aware o any non-trval results establshng ash Equlbra or -person red-and-black games. Ths paper consders a -person non-constant sum red-and-black game, n whch we have players and a gamblng house, that at each stage o the game has a postve probablty o wnnng the players bets. Secton 2 ntroduces the notaton and the termnology we wll use n the ollowng sectons and sets up the ormulaton o the non-constant sum redand-black game. In Secton 3, we show that the wn probablty uncton s super-addtve n 1 s n 1 s and t satses the condton st st,.e. then t s optmal or all players to bet boldly. Several examples are gven as applcaton o ths result. 2. Prelmnares We magne there are players wth 2 who are engaged n a game played n stages, where they all am to reach a goal ortune g by bettng at each stage an ntegral amount not greater than ther current ortune. However the gamblng house may wn some o the players money. The ntal ortunes o the players orm a vector x 1,...,,..., x, where denotes the postve nteger ortune held by player j, or j 1,...,. Hence, at the ntal state, n the system we have a total amount o money equal to M j1. We assume that g M < 2g. Durng the game the amount o money held by the players can become less than M at some stage o the game all o the players lose and the money goes to the gamblng house. The state space S o ths game s dened as the set o all possble ortunes held by the players: S {x 1,..., x : 0 g or j 1,..., }. Ths -dmensonal state space s dscrete and t ncludes several absorbng states. In act, ater one o the players reaches the goal g, or the total amount o money held by the players becomes less than g, the game does not move to any other state. At each stage, each player bds an ntegral amount o money, whch must be less than or equal to the ortune that he holds. We assume each player chooses hs acton wthout any knowledge about the actons chosen by the other players. In act, all the games we consder are noncooperatve games.

3 on-constant sum red-and-black 3 Suppose that player j bds an amount o money, where hs acton set s { {1,..., xj } x A j x 1,..., x j {1,..., g 1} {0} {0, g} In our game we requre each player to bd at least one unt o money he has a postve ortune less than g;.e. 1 > 0, or any j 1,...,. I we allow all o them to bd 0, then or ths game there s a trval, unnterestng ash Equlbrum, n whch all players bd 0 orever, and the game stays n the same state perpetually. The payo uncton or each player s gven by an ndcator uncton, whch takes value 1 the player reaches the goal g, and 0 otherwse. otce that n ths game the total amount o money n the system can stay the same as the orgnal amount M or eventually decrease at some stage o the game the gamblng house wns the players bets. Hence, only one o the players can reach the goal, or the sum o the players ortunes becomes less than g none o them wll be able to reach t. The sum o the payos assocated to each set o strateges s not constant.e. t can be 0 or 1, so the games descrbed here are non-constant sum stochastc games. otce that at each stage o the game the players have a postve probablty o losng at least one unt o money. 3. on-constant sum -person red and black game Suppose that at stage m each player j has 1 x m j g 1 unts o money, and he bds an amount 1 a m j x m j, or j 1,...,. The law o moton or the game at stage m s dened by: x m+1 1,..., x m+1 x m 1 a 1,..., x m a w.p. 1 j1 aj, 1 a x m a,..., x m a w.p. a1, 1 a x m 1 a 1,..., x m j + a k,..., x m a w.p. aj, 1 a x m 1 a 1,..., x m a w.p. a. 1 a 1 where the uncton : [0, 1] [0, 1] s an ncreasng, contnuous nonzero uncton wth 0 0 and s s, whch represents the wn probablty

4 4 LAURA POTIGGIA uncton o the game. We notce that j1 xm j < g at some stage m, then the probablty o reachng the goal g s equal to zero or all the players, hence the game stops. Also ater one o the players reaches the goal g, he stops stakng money, hence the game stops, and the probablty or the other players to reach ther goal becomes zero. In concluson, n our games t s mpossble or more than one player to reach the goal because o our assumpton that the total amount o money M s smaller than 2g. Snce s s, we see that or any j 1,..., [ ] E[x m+1 j x m j ] x m j a m a m k j 1 + x m k1 1 am j + a m a m k j 1 am + x m j a m j a m k 1 am [ ] a m x m j a m j j 1 + x m 1 am j + a m a m k j 1 am x m j a m j + a m j 1 am x m j 1 a m and E[x m+1 j x m j ] xm j or x m j 0 or g. Thereore the processes {x m j }, or j 1,...,, o the ortunes o the players are supermartngales. Ths means that the game s subar or all the players. For a non-constant sum -person red-and-black game wth 1 g 1 and g j1 < 2g, or j 1,...,, we can prove the ollowng result. Theorem 3.1. In a non-constant sum -person red-and-black game, assume that the wn probablty uncton s super-addtve.e. s s and t satses the ollowng condton: st st. A ash Equlbrum s or all players to play a bold strategy,.e. x 1,..., x or all j 1,..., and all x 1,..., x S. Proo. Suppose all players adopt a bold strategy and set Q j x 1,..., x P [player j reaches hs goal g when the game starts at x 1,..., x ]. The correspondng law o moton at stage m o the game s gven by:

5 on-constant sum red-and-black 5 x m+1 1,..., x m+1 0,..., 0 w.p. 1 j1 xj,,..., 0 w.p. x1, ,..., w.p. x. 2 Thereore the expected return to player j when all players use a bold strategy s: Q j x 1,...,,..., x Q j 0,..., x,..., 0 1 where Q j 0,...,,..., 0 1, snce g. In order to prove that or player j a bold strategy s optmal when all other players play boldly, t suces to show that Q j. s excessve see Theorem 3.10 Matra and Sudderth [3]. Hence, we need to show that the expected return that player j gets by ntally stakng an amount 1 and then playng boldly or the rest o the game, s always less than or equal to the expected return he gets playng boldly rom the begnnng. Suppose now that the rst stage o the game player j stakes an amount 1. Hs expected return n ths case s gven by: σ j x 1,...,,..., x j x x + j x x 1 + j x Q j 0,..., Q j + j x, 0,...,,..., 0+ x,..., Q j 0,...,,..., + j x +

6 6 LAURA POTIGGIA + 1 where Q j + j + j x Q j 0,..., j x,..., 0 1, 1 x + j x snce Q j 0,...,,..., 0 x g, 1 Q j 0,...,,..., 0 0, snce < g, x, 0,...,,..., 0 Q j 0,...,,..., + j Thereore we have σ j x 1,..., x + x k j x + 1 x + j x. In order to prove that Q j. s excessve we need to show that the ollowng nequalty holds: Q j x 1,..., x σ j x 1,..., x, 3 whch s equvalent to: 1 x + x k j x + 1 x + j x. In order to show that nequalty 3 holds we need to decompose the quantty as ollows. x 1 x. + x k + x k + x k + x k x k + x k + x k a j + x k + x k + x k + x k + x k + x k + x k + x k

7 on-constant sum red-and-black 7 Suppose the wn probablty uncton s super-addtve,.e. s s, and t satses the condton st st, or any s, t [0, 1], hence we can prove the ollowng result. + x k + + x + k + x k + x k + x k x k + x k x k + x k In concluson nequalty 3 holds or any 1. Thereore we proved that Q j. s excessve or player j and that the prole Bold,..., Bold s a ash equlbrum or ths class o games. Remarks 3.1. The ormulaton o ths game come rom the dea that can be consdered as a penalzng uncton mposed to the players. For example we can magne a stuaton where there s a state agency or the same gamblng house that has the power to control the players by penalzng ther probablty o wnnng. Example 3.1. We consder the wn probablty uncton s ws wth 0 < w < 1, ntroduced n Pontgga [5] or the weghted red-and-black game. It s clear that s s or all s 0, 1 and that ths uncton s super-addtve, s + t ws + wt ws + t s + t. Also t s easy to show that ths uncton satses the condton st wswt w 2 st wst st. Hence t ollows rom Theorem 3.1 that Bold,..., Bold s a ash Equlbrum or a non-constant sum -person weghted red-and-black game. Example 3.2. We consder the wn probablty uncton s s p or some p 1, ntroduced n Chen and Hsau [1]. It s easy to show that s s and that the uncton s super-addtve, snce s+t s p +t p s+t p s+ t. Also the second condton s satsed snce st s p t p st p st. Hence, t ollows rom Theorem 3.1 that Bold,..., Bold s a ash Equlbrum or a non-constant sum -person red-and-black game wth ths wn probablty uncton. Acknowledgements I would lke to thank Dr. Wllam Sudderth or hs suggestons and gudance, and the reeree or hs/her very helpul comments and eedback.

8 8 LAURA POTIGGIA Reerences [1] Chen, M. and Hsau, S Two-person red-and-black games wth bet-dependent wn probablty unctons. Journal o Appled Probablty, 43, [2] Dubns, L. and Savage, L Inequaltes or Stochastc Processes: How to Gamble You Must, 2nd edn. Dover, ew York. [3] Matra, A.P. and Sudderth, W.D Dscrete Gamblng and Stochastc Games. Sprnger, ew York. [4] Pontgga, L Topcs n Stochastc Games. Doctoral thess, Unversty o Mnnesota, Mnneapols. [5] Pontgga, L Two-person red-and-black wth bet-dependent wn probabltes. Advanced Appled Probablty, 37, [6] Ross, S.M Dynamc Programmng and Gamblng Models. Advanced Appled Probablty 6, [7] Secch, P Two-person red-and-black stochastc games. Journal o Appled Probablty 34,

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